Mr Kurz is on course to become the world's youngest national leader.
The People's Party was set to win more than 31%. It is so far unclear whether the Social Democrats or the far-right Freedom Party will finish second.
Short of a majority, Mr Kurz's party could seek an alliance with the anti-immigration Freedom Party.
Addressing his supporters, Mr Kurz said: "It is time for change in this country. Today is a strong order for us, to change this country, and I say thank you to you all who made this possible.
"I'm overwhelmed, I am happy, and I look forward to working for Austria."
Before the election, Mr Kurz served as Europe's youngest-ever foreign minister, after he was appointed in 2013 aged just 27.